Three-dimensional unsteady transonic flow: An integral equation formulation

ABSTRACTGreen's functions have been a tool to analyze important reservoir geomechanics effects like subsidence, compaction, well closure, and permeability variation. However, the classical Green's function is related to linear elastic, homogenous, and geometrically simplified media. The use of auxiliary functions that consider heterogeneous media has the potential to simplify the numerical analyses of some complex problems. This research work proposes a new approach to Green's function method based on artificial neural networks (ANNs) to consider stratified media with direct application in reservoir geomechanics. These solutions obtained through ANN have as output the normalized displacement at any point of the stratified massif due to the application of a point load inside the massif. The input data of the proposed ANN are the material properties of the media, the point load, and the point of interest. By using the developed ANN‐based Green's function together with the classical Green's function approach, it is possible to obtain individually the displacement at any point of the massif due to a pore pressure variation within the reservoir, only discretizing the reservoir boundary. This leads to a more efficient method when compared to traditional methods, such as the finite element method. For the points of interest, seafloor, and top and bottom of the reservoir, the numerical example presented was 25 times faster than the classic Green function approach on a computer with 12 threads of 2.6 GHz and 32 GB RAM. This reduction in processing time is crucial for decision‐makers to act in field applications.

The green's function method in quantum statistics

Simulations that are meant to determine the steady-state distribution of a diffusible solute such as oxygen in tissues have typically used finite difference methods to solve the diffusion equation. Finite difference methods require a tissue mesh with enough points to resolve oxygen gradients near and between discrete blood vessels. The large number of points that are typically required can make these calculations very slow. In this paper, we investigate a numerical method known as the Green's function method which is not bound by the same constraint. The Green's function method is expected to yield an accurate oxygen distribution more quickly by requiring fewer mesh points. Both methods were applied to calculate the steady state oxygen distribution in a model simulation region. When the Green's function calculation used meshes with 1/2, 1/4 and, 1/8 of the resolution required for the finite-difference mesh, there was good agreement with the finite difference calculation in all cases. When the volume of the domain was increased 8-fold the Green's function method was able to calculate the O2 field in 22 minutes, whereas the finite difference calculation is expected to take approximately 1 week. The number of steps required for the Green's function calculation increases quadratically with the number of points in the tissue mesh. As a result, small meshes are calculated very quickly using Green's functions, while for larger mesh sizes this method experiences a significant decrease in efficiency.

A novel surface integral equation (IE) formulation for magneto-quasi-static analysis of current flow in two-dimensional conductors situated in lossy layered substrates is proposed. Traditional approach is based on the volume IE formulated with respect to the unknown current distribution in the conductors cross-section. In this work, we transform the volume IE to a surface IE via representation of the volumetric current density as a superposition of the waves emanating from the conductors boundaries. The resulting surface IE features a single unknown surface current density unlike the traditional surface IEs which require both electric and magnetic surface current densities. Also, the resultant single-source surface IE formulation involves only the electric field Greens functions instead of both electric and magnetic field Greens functions as required by the traditional surface IE formulations. The latter property makes the proposed IE formulation particularly suitable for analysis of current flow in conductors embedded in lossy layered substrates.

The elastic response of a vacancy in a semi-infinite fcc copper lattice containing a free surface is calculated by using a new multiscale Green's function method. The method treats the lattice distortion near the vacancy at the atomistic level and the free surface at the macroscopic continuum level in the same formalism. The lattice is modeled using the lattice statics Green's function that fully accounts for the discrete atomistic structure of the lattice and can model a large crystallite containing a million atoms without excessive CPU effort. The method is especially useful for modeling the elastic response of nanocrystals containing point defects in which surfaces and interfaces play important roles. The method bridges the length scales seamlessly by relating the microscopic lattice distortion near a point defect to measurable macroscopic parameters of the solid such as the strain and the displacement field at a free surface. Using the interatomic potential derived by Cleri and Rosato, the lattice distortion, relaxation energy, and relaxation volume due to a vacancy are calculated in an otherwise perfect copper lattice for a million-atom model containing a free (100) surface. The calculated value of the relaxation volume is in excellent agreement with the observed value. Numerical results are also presented for the strain and the displacement fields at the free surface due to a vacancy and the interaction energy between a vacancy and the free surface in anisotropic semi-infinite copper.

Resonance plays a key role in the formation of many physical phenomena. The complex-scaled Green's function method provides a powerful tool for exploring resonance. In this paper, we combine this method with the theory describing deformed nuclei with the formalism presented. Taking $^{45}\text{S}$ as an example, we elaborate numerical details and demonstrate how to determine the resonance parameters. The results are compared with those obtained by the complex scaling method and the coupled-channel method and satisfactory agreement is obtained. In particular, the present scheme focuses on the advantages of the complex scaling method and the Green's function method and is more suitable for the exploration of resonance.

The formulation of the Green's-function method, recently suggested by Ziman, for solving the energy-band problem in a so-called "muffin-tin" potential (zero outside spheres representing the various atoms, and spherically symmetrical within each sphere) is transformed to a form very much like that of the Augmented-plane-wave (APW) method, but cannot be transformed exactly into the APW formulation. The wave function set up by the Green's-function method is a superposition of plane waves satisfying the wave equation inside as well as outside the spheres representing the atoms, having a discontinuity of slope at the surfaces of the atomic spheres, enough to produce the correct solution of the Schr\"odinger problem outside the spheres. To get correct wave functions inside the atoms, one would have to construct augmented plane waves from these plane waves.

We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Green's function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...

The fracture behaviors of nanoscale sp-bonded materials have been studied using the molecular dynamics and lattice Green's function methods. The initial atomic structures of the crack are determined both from the elastic solutions as well as from those by lattice Green's function method for the infinite systems. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation, appropriate for absolute zero temperature, After the lattice Green functions of the absolute zero temperature have been determined, the lattice parameters and interatomic force constants are adjusted to fit to materials at temperature T. In general, we have found that the lattice trapping and stress intensity factors for dislocation emission K Ile , The fracture and strength properties are also investigated for the nanocrystailine materials like semiconductor quantum wire and nanotubes. The O(N) tight-binding molecular dynamics (TBMD) method is used to analyze the reconstruction of atomic bonding near the crack tip as well as the cleaved surface. We compare the fracture behavior of nanoscale materials with those of corresponding bulk-size materials.

<sec><p indent="0mm">Exotic nuclei far from the β stability line have become important scientific goals in the studies of experimental physics at large scientific facilities and theoretical research due to the rich new physics. In these exotic nuclei, the neutron or proton Fermi level is close to the continuum threshold. The pairing correlation could scatter the valence neutrons or protons into the continuum. This leads to the extended neutron or proton density distributions in these exotic nuclei. Therefore, properly describing the pairing correlation and continuum is crucial for studying the structures and properties of these exotic nuclei. The Hartree-Fock-Bogoliubov (HFB) theory is one of the promising tools for describing exotic nuclei. In this theory, people usually solve the HFB equation directly in the coordinate space or the Woods-Saxon basis. In the coordinate space, when employing box boundary conditions to discretize the continuum, the behavior of the wave functions at the boundaries will be affected by the size of the box. Meanwhile, one could not obtain the energy and width of the resonant states directly from the discretized continuum states. The Green’s function (GF) is a simple and effective tool for handling the continuum, which has been widely used in nuclear structure research. The GF can be established by using the wave functions that satisfy the equation of motion and the proper bound boundary conditions for the bound and continuum states. Then one can use the loop integral of the GF on the complex energy plane to construct the particle density and level density. In this way, the continuum states can be included in the density with proper boundary conditions, and thus the extended density distribution can be properly described. Furthermore, the GF on the complex energy plane and the level density can be used to identify the resonant states directly. </sec><sec> This paper briefly reviews the development of the GF method in the continuum density functional theory, focusing on its application in describing halo phenomena and single-particle resonance states in exotic nuclei. For the description of the halo phenomena, this review takes the neutron-rich Zr as examples to show the results given by Skyrme HFB calculations with the GF method. The extended neutron particle and pair density distributions can be more properly described by the GF method, compared to the box-discretized method. The self-consistency to deal with the pairing correlation and the continuum is important to determine the asymptotic neutron particle and pair density distributions. Furthermore, with the level density obtained by the loop integral of the GF on the complex energy plane, one can describe both the bound and resonant states on the same footing. Recently, in the relativistic mean field theory, a new method to obtain precise information for the resonant energy and width is proposed by using directly the poles of the GF on the complex energy plane. This helps explain the conservation and breaking of pseudospin symmetry in the nucleon single-particle levels. Additionally, Green’s function method is easily compatible with various theoretical models, thus in the future it can be further applied to describe the exotic nuclear collective resonances and nuclear reaction processes. </sec>

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

Ab initio green's function calculations on highly conducting polymers: Effects of electron correlation and aperiodicity

The Modified Global Green's Function Method (MGGFM) is an integral technique that is characterized by good accuracy in the evaluation of boundary fluxes. This method uses only projections of the Green's Function for the solution of the discrete problem and this is the origin of the term 'Modified' of its name. In this paper the local strategy for calculating the projections of Green's function using de Finite Element Method (FEM) are detailed. The numerical examples show some aspects of the method that had not yet been observed and good results for the flux in all nodes of the mesh.

This paper, on one hand, is an introduction to thermo‐field dynamics (TFD), on the other hand a derivation of some interesting features of Green's functions in the frame of thermo‐field dynamics. It is proved that in the language of TFD, Green's functions at finite temperature take a very simple form. Through the applications of TFD to many‐body systems it is shown that TFD is a straightforward method in dealing with finite temperature many‐body systems. As an application of Green's function method in TFD, we derive a Kogan‐type formula for the non‐equilibrium dynamic conductivity or laser‐pulse heated condensed matters, where electrons and ions are held at different temperatures.

A new empirical energy-band parametrization scheme based on the Green's-function method has been developed and was applied to Cu and Ag. The scheme utilizes the logarithmic derivatives associated with an ab initio muffin-tin potential ${V}^{(0)}(r)$. The scheme can be understood in terms of the addition to ${V}^{(0)}$ of $E$- and $l$ -dependent square-well potentials the depths of which ${\ensuremath{\nu}}_{l}(E)$ are adjusted to yield the correct (empirical) energy bands. The ${\ensuremath{\nu}}_{l}(E)$ are found to be smooth functions of $E$ which can be accurately approximated by low-order polynomials. An accurate fit for $d$ -band metals over a roughly 1-Ry range requires only seven adjustable parameters\char22{} a number smaller than required by other schemes. Extensive tests of the approach using results of first-principles calculations were carried out in precisely the same manner as proposed for the empirical application, and the results indicate that this scheme is more accurate than other approaches using more parameters. The seven pieces of data used in the empirical parametrization for Cu and Ag were the $s$, $p$, and $d$ phase shifts required to fit the Fermi-surface geometry and four firmly identified vertical energy gaps: ${E}_{F}\ensuremath{-}{X}_{5}$, $X_{4}^{}{}_{}{}^{\ensuremath{'}}\ensuremath{-}{X}_{5}$, ${X}_{5}\ensuremath{-}{X}_{3}$, and ${L}_{1}^{u}\ensuremath{-}L_{2}^{}{}_{}{}^{\ensuremath{'}}$. The empirical ${E}_{n}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ were obtained for a large range of ${E}_{F}$ values (relative to the constant part of the potential). Except for the rather high energy levels (e.g., the upper $X$, $W$, and $K$ states) the relative band structures prove to be rather insensitive to the ${E}_{F}$ value. The presently available and firmly established data do not narrow the permissible range of ${E}_{F}$. Comparisons with several recent Cu and Ag calculations show that the present bands are in better accord with experiments. The empirical ${E}_{n}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$, which are required to fit the input data, are also found to agree within experimental uncertainties with all additional data related to level positions. Cu, for which more data are available, is particularly well checked. To obtain some information about the effective interactions ${V}_{l}(E,r)$ and the associated wave functions, a set of coupled integro-differential equations is derived which relates these quantities to the ${\ensuremath{\nu}}_{l}(E)$. However, it appears that the solutions to these equations are not unique unless some constraints are imposed on the correction ${V}_{l}(E,r)\ensuremath{-}{V}^{(0)}(r)$. A suggestion is made for obtaining approximate useful wave functions prior to the resolution of the nonuniqueness problem. From our experiences and a consideration of the merits of the new scheme, it is evident that it should be very useful in the study of the electronic structure of various solids.